Bifurcation and Stability of Two-Dimensional Activator–Inhibitor Model with Fractional-Order Derivative

In organisms’ bodies, the activities of enzymes can be catalyzed or inhibited by some inorganic and organic compounds. The interaction between these compounds is successfully described mathematics. main purpose this article to investigate dynamics activator–inhibitor system (Gierer–Meinhardt system), which utilized describe interactions chemical biological phenomena. considered with a fractional-order derivative, converted an ordinary derivative using definition conformable fractional derivative. obtained differential equations are solved separation variables. stability positive equilibrium point analyzed discussed. We find that locally asymptotically stable, source, saddle, non-hyperbolic under certain conditions. Moreover, concentrates on exploring Neimark–Sacker bifurcation period-doubling bifurcation. Then, we present numerical computations verify theoretical results. findings work show governing undergoes These types occur in small domains, as shown theoretically numerically. Some 2D figures illustrated visualize behavior solutions domains.